You cannot use the APU because you will not know if it is on fire, not a good situation.
If you lose both loops of the fire detection system on an engine, you still have 149 fire detectors in the back of the aircraft, half sitting on the left, half sitting on the right.
As far as shutting the engine down, after learning that it is on fire I would be inclined to execute the engine fire, severe damage or separation checklist recall items. That should tick that box.